Master's Thesis Morphisms in Logic, Topology, and Formal Concept Analysis by Markus Krötzsch Overseeing Professor: Prof. Dr. Steffen Hölldobler, TU Dresden Supervisor: Dr. Pascal Hitzler, TU Dresden/University of Karlsruhe External Supervisor: Prof. Dr. Guo-Qiang Zhang, CWRU Cleveland International Center for Computational Logic Department of Computer Science Dresden University of Technology Dresden, February 2005 Copyright notice This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/ by-nc-sa/2.0 or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, Cal- ifornia 94305, USA. Also see http://science.creativecommons.org for further explanation. Abstract The general topic of this thesis is the investigation of various notions of morphisms between logical deductive systems, motivated by the intuition that additional (cat- egorical) structure is needed to model the interrelations of formal specifications. This general task necessarily involves considerations in various mathematical dis- ciplines, some of which might be interesting in their own right and which can be read independently. To find suitable morphisms, we review the relationships of formal logic, al- gebra, topology, domain theory, and formal concept analysis (FCA). This leads to a rather complete exposition of the representation theory of algebraic lattices, including some novel interpretations in terms of FCA and an explicit proof of the cartestian closedness of the emerging category. It also introduces the main con- cepts of “domain theory in logical form” for a particularly simple example. In order to incorporate morphisms from FCA, we embark on the study of various context morphisms and their relationships. The discovered connections are summarized in a hierarchy of context morphisms, which includes dual bonds, scale measures, and infomorphisms. Finally, we employ the well-known means of Stone duality to unify the topo- logical and the FCA-based approach. A notion of logical consequence relation with a suggestive proof theoretical reading is introduced as a morphism between deductive systems, and special instances of these relations are identified with mor- phisms from topology, FCA, and lattice theory. Especially, scale measures are rec- ognized as topologically continuous mappings, and infomorphisms are identified both with coherent maps and with Lindenbaum algebra homomorphisms. Acknowledgements Parts of this work have been funded by the German Academic Exchange Service (DAAD), the Gesellschaft von Freunden und Förderern der TU Dresden, e.V., and Case Western Reserve University, Cleveland; their support is gratefully acknowl- edged. This work could not have arrived at its current form without the support of my supervisors Dr. Pascal Hitzler, Prof. Steffen Hölldobler, and Prof. Guo-Qiang Zhang, who provided me with all the freedom and time I could have wished for. I warmly thank GQ for inspirations, good advice, and hospitality during my time in Cleveland, where most of Chapter 3 was written. With respect to Chapter 4, I am indepted to Prof. Bernhard Ganter for giving helpful hints, especially to his manuscript [Gan04] that greatly inspired this work. Furthermore, I would like to thank Grit Malik for helpful discussions on this topic. Although many people contributed to my academic education, the knowledge that enabled me to write this thesis largely goes back to two people: Pascal Hitzler and Matthias Wendt. My thanks to Pascal cannot possibly account for his influ- ence on my studies, which traces back to my first contact with formal logic in undergraduate courses. Over the years, he provided me with numerous opportu- nities, hints, discussions, and an inexhaustible optimism that was often a major source of my motivation. The discussions with Matthias have been extremely inspiring, though I was usually content to follow his ideas – at least in parts. My understanding of alge- braic semantics, Stone duality, and also Logic in general, mainly goes back to this influence. I regret that, now that I come to comprehend some of these topics, he is already concerned with new subjects beyond my current mathematical horizon. Further academic and personal thanks are given to Sebastian Bader, Matthias Fichtner, Christian Kissig, Loïc Royer, Prof. Michael Thielscher, and the mem- bers of the Flux group, for inspiring discussions, good advice, and a nice time. Ai(mée) Liu, Mary Sims, Amit Sinha, Jacek Szymanski and Ivan Vlahov con- tributed a lot to my well-being during my stay in Cleveland. I also thank the professors of the Computational Logic programme, in partic- ular Prof. Franz Baader, Prof. Horst Reichel, Prof. Steffen Hölldobler, and Prof. Michael Thielscher, for their support of the rather unconventional organization of my studies. Special thanks are due to my family and friends, who far too often have been neglected over my recent work, and to whom I have hardly been able to really explain the contents of my studies. I thank my parents for their help and support – their contribution to this work is immeasurable. Last and most, I thank Anja for bearing with me and my tight working sched- ule, and for all her care and understanding. Contents 1 Introduction 5 2 Preliminaries 13 2.1 Orders and lattices . 13 2.2 Morphisms of partially ordered sets . 17 2.2.1 Galois connections . 17 2.2.2 Closure operators . 20 2.3 Formal concept analysis . 22 2.4 Topology and domain theory . 24 2.4.1 Domain theory . 25 2.4.2 General topology . 27 2.5 Category theory . 29 3 Algebraic Lattices 35 3.1 Algebraic lattices . 35 3.2 Approximable mappings . 39 3.3 A cartesian closed category of formal contexts . 42 3.4 Further representations . 49 3.4.1 Logic and information systems . 49 3.4.2 The Scott topology . 54 3.4.3 Stone duality . 56 3.5 Summary and further results . 60 3.5.1 Further logics . 60 4 Morphisms in FCA 64 4.1 Dual bonds and the direct product . 64 4.2 Continuity for dual bonds . 68 4.3 Functional bonds and scale measures . 71 4.4 Infomorphisms . 77 4.5 A concept lattice of morphisms . 81 4.6 Conclusion and future work . 83 3 Contents 5 Categories of Logics 85 5.1 Logic and FCA . 86 5.2 Consequence relations . 89 5.3 Continuous functions . 94 5.4 Infomorphisms and coherent maps . 97 5.5 Future work . 99 Bibliography 101 List of Symbols 106 Index 108 4 Chapter 1 Introduction In Computer Science, formal logics generally are perceived as a tool for specifi- cation and reasoning, where the latter – partly due to the efforts of Proof Theory – is often identified with a process of computation. This intuition turns out to be feasible for many logical formalisms, and today numerous concrete implementa- tions of reasoning mechanisms are available. Classically, such implementations are the domain of logic programming [Llo87], but growing demands lead to de- velopments in other areas as well. Most recently, ontology research opened up new applications for knowledge representation and reasoning, and gave rise to novel logic-based reasoning formalisms, such as F-Logic [KLW95] or Descrip- tion Logic [BCM+03]. Many more approaches, both theoretical and practical in nature, engaged in similar efforts to provide means of specification and reasoning for some particular application area. However, in most cases, “specification and reasoning” restricts to the specification of and the reasoning on top of some particular deductive sys- tem (i.e. logic program, ontology, . ). What is often neglected is the question of how to specify the relationships between such deductive systems and how to infer consequences for such interrelations. Nonetheless this question appears to be vital for the success of some – probably most – of the targeted applications of formal logics. On the one hand, use-cases of practical dimensions can hardly be based on a single huge specification, but will rather require modularization into numerous smaller ones. Reasoning in such a setting clearly requires the specification not only of the modules themselves, but also of the exact relationships between them. On the other hand, situations with even higher levels of heterogeneity naturally occur in ontology research, e.g. in the context of a semantic web. There, one faces a scenery of multiple distributed specifications which may not even use a common logical language, and which have not been conceived as modules of some over- all deductive system. This situation represents a considerable challenge to current research, and neither theoretical nor practical approaches to this problem are de- 5 Introduction veloped to a satisfactory extent. Given the amount and diversity of available logical formalisms, one obviously cannot expect this problem to have a simple solution. In fact, the first question that arises is how to specify the aforementioned “relationships” between deduc- tive systems at all. Initially, one is faced with a mere collection of specifications, lacking additional structure that could be used for interrelating them. A priori, it is not clear how this additional structure should look like, and indeed there might be various reasonable choices, strongly depending on the particular kind of logi- cal formalisms that are to be taken into account. However, the primitive concepts of such investigations most certainly are the relationships between a single pair of specifications. In ontology research, such relationships are sometimes called ontology mappings [KS03]. In this generality, this notion does not yield a lot of structural information, and we therefore make the additional assumption that rela- tionships between specifications have a direction. This can be justified on practical grounds as well, since relationships between specifications often come with a pre- ferred direction for the flow of information. Examples include modules which are to be included into some bigger specification, and ontologies that have been gath- ered from the semantic web to be processed in (the deductive system of) a local reasoner.